## Chapter 13 — Three of twelve counts conjunctions ### Lesson map Lesson 13. Map, example, EX059. ```mermaid flowchart TD C["46 ones inside 60 cells"] --> R["A cell rate"] U["Nine laptops with at least one zero"] --> W["A row rate"] R --> N{"Are 46/60 and 3/12 the same claim?"} W --> N N -->|No| K["Report the reduction you computed"] ``` Cell totals and row totals answer different questions. Their failure sets overlap. ### Logical drawing ```text pass all five FB01, FB08, FB12 update zeros FB03, FB07 encryption zeros FB02, FB06, FB07, FB11 standard-user zeros FB02, FB04, FB07, FB09 lock zeros FB05 backup zeros FB03, FB07, FB10 union of those zeros nine keys, not fourteen ``` ### Learning outcomes After this chapter, you should be able to: 1. Compute the cell total, the conjunction, and the union from the dictionary. 2. Explain why fourteen failure-cells are not fourteen laptops. 3. Refuse a kappa, an average disguised as the all-five count, or a tie-break this run did not need. ### 13.1 Overlapping failure sets are why the rates do not add. The five printed numerators are 10, 8, 8, 11, and 9. Their sum is 46. Each laptop contributes five cells, and twelve laptops contribute 60 cells. So 46 of 60 cells are `1`. That ratio is a property of cells. It is not `3/12`. The all-five count asks a harder question of each row: are all five cells `1`? Only `FB01`, `FB08`, and `FB12` meet it. A row with a single `0` contributes four, or even five, to a careless reading of "mostly fine," and contributes nothing to `3/12`. Both arithmetic results are in the object. Substituting one for the other changes the question. The failure side shows why adding the column gaps over-counts laptops. Update fails on `{FB03, FB07}`. Encryption fails on `{FB02, FB06, FB07, FB11}`. The standard-user cell fails on `{FB02, FB04, FB07, FB09}`. The lock fails on `{FB05}`. Backup fails on `{FB03, FB07, FB10}`. Those sets contain 2, 4, 4, 1, and 3 keys, and 2+4+4+1+3 is 14. The union contains nine keys: `FB02`, `FB03`, `FB04`, `FB05`, `FB06`, `FB07`, `FB09`, `FB10`, and `FB11`. Nine plus the three clean rows is twelve. The inequality is ordinary. A sum of sizes equals the size of the union only when the sets are disjoint. They are not. `FB07` sits in four of the five failure sets and only passes the lock, so its five-cell sum is 1. `FB02` fails encryption and the standard-user cell. `FB03` fails updates and backup. Adding 14 and announcing "14 laptops failed" invents two laptops and hides the overlap. `min` selects `FB07` because 1 is the unique minimum. Python dictionaries preserve insertion order as a language guarantee from 3.7 onward, yet a tie never arose, so that rule stayed idle. Landis and Koch define kappa for agreement between observers on categorical labels. One person filled this matrix. A second observer was not used, and a kappa was not computed. Do not decorate 46 of 60 with a coefficient the study did not run. The later narrative in which twelve of twelve pass is still a different reading. It is not a reinterpretation of these nine keys. ### 13.2 Worked example | Quantity | Value from this dictionary | Decision | |---|---|---| | Yes-cells | 46 of 60 | A cell total | | All-five rows | 3 of 12 | A conjunction | | Sum of column-failure sizes | 14 | Not a laptop count | | Union of failing keys | 9 | The rows that miss the conjunction | > **Synthetic case.** The sets are the zeros already visible in the embedded lists. ### Errors and limits An average of the five column rates is still not the conjunction. Overlap is the reason the story feels inconsistent until the sets are written down. ### Apply - **EX059 — Which total?** A summary says "about three quarters pass, so 9 of 12 pass all five," and it adds a kappa. What does the dictionary actually support? *Rubric:* 3 = 46 of 60 is the cell total; 3 of 12 is the conjunction on FB01, FB08 and FB12; column-failure sizes sum to 14 across a union of nine keys; FB07's sum is 1 and there is no tie; no kappa is computed;; 2 = the cell total and the row total stay distinct;; 1 = one total;; 0 = none. ### Continue studying — lesson 13 (chapter 13 of 20) [← Chapter 12](#ch-12) · [Course chapter index](../SUBJECTS.md) · [Chapter 14 →](#ch-14) Continue to the next authored lesson. ## Worked answers **EX059.** 46 of 60 is the cell total. 3 of 12 is the conjunction on FB01, FB08 and FB12. Failure sizes sum to 14 across nine keys. FB07's sum is 1, with no tie. No kappa is computed.